Here is a statistical summary:

            marks
count  106.000000
mean    73.509434
std     12.449705
min      0.000000
25%     66.250000
50%     76.000000
75%     80.000000
max     95.000000

You can see the distribution of marks here:

Overall, this cohort produced a strong and diverse set of projects, with a mean mark of 73.5 and a median of 76 reflecting a high general standard of work. Many projects demonstrated clear understanding of core game-theoretic ideas, and a substantial number went beyond basic applications to explore evolutionary dynamics, repeated games, learning, and simulation in creative and engaging ways. The upper quartile of marks (80+) contained several genuinely impressive pieces of work, showing depth of analysis, originality, and excellent integration of theory, computation, and interpretation.

Across the cohort, the most successful projects shared a few common features: a clearly stated mathematical model, well-explained simulations, and results that were carefully interpreted rather than simply presented. Strong projects also showed awareness of the distinction between static and dynamic solution concepts, and used computational tools to extend insight rather than replace explanation. Presentations were often a strength, with many students communicating complex ideas clearly and confidently, even when the written paper was more limited.

The main areas for improvement were also consistent across projects. A number of submissions lacked sufficient detail about what was actually done, making results difficult to interpret or reproduce. In some cases, modelling choices were not clearly justified, or game-theoretic concepts were invoked without being formally defined or correctly applied. Several projects would have benefited from fewer ideas executed more carefully, clearer visualisation of results, and tighter alignment with the methods taught in the course. Overall, this was a very strong cohort, and the distribution of marks reflects both the high quality of the best work and the clear learning progress made across the group.

I will no longer have a research project component in this course in future years and feel that this was a really nice set of projects to end on. Thank you for your hard work.

If you’re interested you can find a list of all the past projects at the assessment page.

I believe I have written detailed feedback on each of your projects but again: I welcome any conversation you would like to have so please get in touch with me.

Thanks again for all the effort you put into this assessment. For those of you graduating I wish you all the best.

• Matching games
• Subgame perfection

You can find the board I wrote on here

I worked through the Gale Shapley algorithm for the examples in that pdf and then I went over the solutions to exercise 3 from the subgame perfection chapter of the course.

You can see the recordings here:

• From Friday
• From Today

You can find this example and the solutions in the exercises section of the Game Theory notes.

You can find a recording of the class here

We finished the class by discussing my notes on principles for presentation: https://vknight.org/pop/.

One group from previous years put their presentation on YouTube and you can find it here.

You can see a recording of this here.

You can find a recording of the class here.

Below is a summary of my recollection of our conversation:

• The paper should:
  • Say what you have done.
  • Say how you did it.
  • Say what you found.
  • Say why (answer “so what?”).
• We spoke about whether or not to use visualisation in a paper. Without this being a rule I’d suggest it is a good idea when appropriate.
• We spoke about the need to include reference (in the 2 pages).
• We spoke about code and where to include it.
• We spoke about levels of depth (and how specifically the example paper had little depth).

I asked you as a class to mark the project, based on the paper alone: you came to an average of about 61/100.

I threw it in to Chat GPT and it gave it a mark of 52/100.

We will come to the mark that I feel it warrants but only after discussing the presentation.

You can see a recording of this here.

Social Choice

For the purpose of studying social choice I asked you all to vote on which topic from game theory you wanted us to revise on Monday.

Using the first past the post system: the Moran Process got 11 votes whilst the other subjects got a total of 19.

Following this I used a different tool which uses more advanced social choice methodology to get your decision. This site asked you to rank all your choices and then gives a decision: it in fact resulted in the same outcome (but this is not necessarily the case). You can see the results here.

Following this I discussed the chapter on social choice highlighting:

• Arrow’s impossibility theorem: there is no perfect system.
• Condorcet’s method which might not always lead to a decision.
• Borda’s scoring method which does always lead to a decision.

Shapley Value

I then asked for 3 volunteers to try and land paper plans in a target.

You can find the results of this in the blue and red part of the so called characteristic function game.

You can find the calculations we made on the board here:

(Note that the value of Tom and Franks coalition in the blue game should be 6.)

At the very end of the class, after we computed the Shapley value for the red game Tom asked an excellent question:

Since I can get 3 alone and the Shapley value gets me 17/11 why would I act as part of the coalition?

I made a mistake here, saying that something must be wrong.

In fact the game is not super additive and so the Shapley value here does not guarantee so called individual rationality.

Tom suggested a change to the characteristic function that would ensure it is super additive that would ensure this.

You can find a Jupyter notebook here with code to carry out the calculations for each other these scenarios.

You can see a recording of this:

• here for matching games.
• here for auctions.

Matching Games

For Matching games I started by asking you to come up with individual rankings for physicists to work with mathematicians and versa:

Using this we described potential matchings. In our particular case this always lead to a bad matching for Turing but this was nonetheless the only matching that was stable.

We then spoke about the Gale-Shapley algorithm which is a powerful yet simple tool for finding stable matchings.

Auctions

Today we started with another auction for a £5 note.

I started by asking you all to write down what you thought was the value of that £5 to you. Some of you assigned quite a high value (more than £5) and some quite a low value (less than £5).

After that I said that the auction would be that the highest bidder would win the £5 and pay the second highest bid.

This lead to a discussion about what the optimal bidding pattern was based on value. Tim, who “won” the £5 (and owes me £7) had a strategy based on the assumption that everyone would bid near £5 so his winning bid of £50 would work.

There are two theorems with the associated chapter, the first of which actually shows in expectation the optimal bid is to in fact bid your value.

You can see a recording of this here.

Pigou’s example is a simple model of congestion there are 2 choices available to traffic, the first is not affected by traffic, the other is heavily affected by traffic (the more people using it, the worse it is).

This simple situation is modelling as a routing game which requires the network diagram and congestion functions you can see here:

The calculations we carried out included:

• Identifying the most efficient flow: half the population take each edge.
• Identifying the Nash flow: the entire population using the shortcut (making it as slow as the other way)
• Finding the Nash flow by optimising the Potential function.
• Finding the optimal flow by finding the Nash flow for a slightly modified game.

The final thing I spoke about was Braess’ Paradox which is an important idea, it is theoretically interesting but empirically has real negative implications: adding capacity to networks may lead to a worse performance.

You can see a recording of this here.

The Moran Process

The Moran process is a game theoretic model of evolution. One of the differences from the Replicator Dynamics equation is that the population is assumed to be finite: so we assumed there is a finite population of \(N\) individuals that can be of any of the types that correspond to actions of the underlying Norma Form Game.

In the example of the Hawk Dove game that we played in class we assumed there were \(N=3\) individuals and the question we attempted to understand was: if we introduce a Hawk in to a population of Doves, what will happen?

The Moran process then follows the following:

1. Calculate the fitness of all individuals: everyone in the population plays the Normal Form game against everyone else. Their utility (or fitness) is given by their type and the type of the individual they play with.
2. Randomly select an individual for copying proportional to their fitness.
3. Randomly select an individual for removal (all individuals are equally likely to be removed).
4. Create a new individual of the same type as the one selected in step 2.
5. Remove the individual selected in step 3.

Repeat that process until there is a single type of individual in the population.

In class we used dice to simulate the above and obtained a probability of 47% of the Hawk taking over.

Here is a notebook with some numeric simulations of the probabilities. If you look through the notes you can see approaches for calculating exact fixation probabilities.

KKT Conditions

After that we spoke about the Karush-Kuhn-Tucker Condition which I described as a rigorous mathematical set of rules for a simple idea of where optima might be on a constrained region.

You can find the notebook I used to draw the function we looked at here.